I agree with your descriptions of the terms, but I think there's often a divergence between empathy (which I find great) and reflecting people's feelings (which I find good with caution). I want people to understand and help each other. But in some situations, reflecting people's feelings encourages them to make poor decisions. I should always provide a space for people to speak without scorn and with understanding, but I don't want to give a false impression of my concerns. Acknowledging that someone's life sucks is subtly different from acknowledging it aloud, and sometimes the subtlety is crucial.

Reflecting peoples feelings is sometimes called "showing sympathy."